Posts Tagged ‘acceleration’

Bounce, part 4

January 2, 2010

Previous parts: 1 2 3

Last time we made progress on figuring out how high a tennis ball can bounce in the classic experiment where we drop the tennis ball on top a basketball. We didn’t find the answer, but we said that if the tennis ball picks up a speed v in falling, then immediately after bouncing off the basketball, it could have a maximum upward speed of 3v.

Today we want to figure out what that means in terms of how high the tennis ball will bounce. It turns out that the tennis ball does not bounce three times as high as it started when it rebounds with three times the speed. In fact it bounces much higher.

After bouncing off the basketball, the tennis ball rises, but slows down under the influence of gravity until it comes to a stop at the top of its trajectory. To understand how high it goes, we must answer the question, “what does the influence of gravity do to the motion of the ball?”

One of the first people to understand this question and its answer was Galileo (although several people came to the correct conclusion before him). We’ll look at a few passages of his famous book, Dialogue Concerning Two New Sciences. (specifically this part)

Galileo begins by stating that he thinks “uniformly accelerated motion”, the motion of a tennis ball thrown into the air, should be very simple.

When, therefore, I observe a stone initially at rest falling from an elevated position and continually acquiring new increments of speed, why should I not believe that such increases take place in a manner which is exceedingly simple and rather obvious to everybody? If now we examine the matter carefully we find no addition or increment more simple than that which repeats itself always in the same manner.

In other words, the way the speed of a falling body changes shouldn’t depend on how high it is, or how long it’s been falling, or how far it’s fallen. It should depend on nothing at all – be always the same.

This may be a lot to swallow, but let’s look at one good reason (not due to Galileo) that we might expect the way gravity acts on a falling object not to change with how high the object is above the Earth’s surface. The radius of the Earth is very large compared to the heights we throw things. We expect that if the effects of Earth’s gravity do change with your distance from the center of the Earth, they ought to do so on a distance scale roughly equal to the radius of the Earth.

That is, if you want a significant difference in the force of gravity, you ought to change your position by something significant compared to the radius of the Earth, since it defines the only natural length scale in this problem. The radius of the Earth is roughly six million meters, so throwing a tennis ball up in the air six meters is completely negligible. We could calculate the effects of gravity using Newton’s gravitational law, but that is unnecessary. Any other reasonable gravity law ought to work out basically the same. Near the surface of the Earth, your height should not affect how gravity acts on you.

This is only one part of what Galileo said. For example, he also believes that how fast an object moves should not affect how gravity acts on it. This belief may have been stimulated by the relativity principle – that all laws of physics should be the same, even when you’re moving. Relativity does not absolutely preclude a force that depends on velocity, though (magnetic forces do this), but velocity-dependent forces are not as simple as velocity-independent forces, and for the time being Galileo is guessing that the way gravity acts ought to be very simple.

We continue with the G-spot’s wise words:

A motion is said to be uniformly accelerated, when starting from rest, it acquires, during equal time-intervals, equal increments of speed.

This is Galileo’s working idea of how things fall. If you drop something, and at the end of one second it goes speed v, then at the end of two seconds it will be going 2*v, and at the end of three seconds 3*v, etc. A plot of speed over time, if we drop an object from rest, should look like this:

The speed of a falling tennis ball, starting from rest.

This plot shows the speed of a falling tennis ball. The tennis ball is dropped from rest, and so starts at speed zero. Gaining equal speed in each moment of time, the speed is directly proportional to time.

Now that we have a theory for what the speed of the tennis ball does, we should be able to figure out how high it goes. The tennis ball reaches its highest height when its speed is zero, so we simply need to keep track of its speed until that speed falls to zero. If we know how fast it was going and for how long, we should also know how far it traveled.

I’ll paraphrase Galileo’s arguments here rather than quoting them, since he does not directly answer our exact question. The relevant pages are 171 – 178.

First, let us suppose it takes the tennis ball a time t to fall before bouncing, and it acquires speed v in that time. We know it bounces back up with speed 3v. It loses speed in the same way it gained speed – the same amount per second. So after a time t, the ball loses speed v, and is down to moving at speed 2v. The ball comes to a stop at the height of its trajectory after a time 3t.

To summarize, if the ball gains and loses the same amount of speed in any moment of time, then if it two balls bounce upward, one three times as fast as the other, the fast one will take three times as long to get reach its apex.

The distance the ball travels just speed * time, which is the green area shaded in the previous drawing.

Here is a plot of the speed of the ball as it rises:

tennis ball's bounce

The tennis ball's return trip. This time it begins going quickly, three times as fast as before, and slows down. It takes three times as long to reach its peak as it took to fall.

It rises three times as long as it fell, and the distance it rises is purple the area in the above chart. Laying the two plots together, we see that the purple area is nine times as large as the green one – three times taller and three times wider.

rise and fall comparison

The green area represents the distance the tennis ball fell (see first figure). The purple area is the area the tennis ball rises after bouncing off the basketball. The tennis ball rises nine times as high as it was dropped from.

Now we have our first answer to how high a tennis ball can bounce when dropped on top a basketball. It can bounce nine times as high, when we make the following assumptions:

  • When things bounce off the ground, they change their direction keeping exactly the same speed and hence bounce back to the same height. (first post).
  • A basketball is so much bigger than a tennis ball that it essentially acts as the ground – the tennis ball bounces off just the same as it would bounce off the ground. (third post)
  • To understand the way something in motion works, we can imagine we are moving alongside it at the same speed so that it isn’t moving from our point of view, and understand it that way. Then we can imagine going back to the frame in which the thing is moving and translating over our new knowledge over. (third post)
  • Gravity pulls an object down such that it gives it the same additional amount of speed in each moment of time. (this post)

My original claim was that I could have understood all these ideas as a child. I think that’s right. I was a pretty bright kid, and if someone had sat down to explain this reasoning to me, and answered my questions, I think I’d have gotten it. But I also hope I’d have realized there’s a problem. When you actually do the experiment, the tennis ball doesn’t bounce nine times as high, or anywhere near that. Three times as high is pretty good for this experiment. So I’d like to think I’d have noticed that, and asked for an explanation of the discrepancy.

We began to discuss this in part two, where we looked at why things bounce to a lower height than they’re dropped from. The assumption about reversing direction and speed when bouncing is simply not correct. It is also not correct to assume that the basketball is so much larger than the tennis ball that it acts like the ground, but this is a smaller source of error. It isn’t true that gravity is completely uniform, either, or that the only influence on the falling ball is from gravity. We’ll look at these things in more detail in a later post.

Before doing that, though, the next post or two will continue looking at the passage from Galileo. This passage isn’t interesting to me simply because it is an early source of someone understanding this fairly simple problem. It’s interesting because it’s an illustration of Galileo laying down a more sophisticated understanding of how we can understand nature. I want to look at what Galileo did and didn’t know, but also at how much he understood about what he did and didn’t know, and how he came to his conclusions.

There’s also a very surprising and egregious logical error in the passage, so we’ll talk about that, too, before returning to the tennis ball a little down the line.


The Renaissance Man Uniform Gravitational Acceleration SMACKDOWN

October 9, 2009

Matt at Built On Facts posts about coriolis forces, and points out that a falling body is deflected by them one eighth as much as one tossed from the ground to the same height, and that they’re deflected in opposite directions. Here’s my attempt to explain this intuitively.

This makes me think of the competing da Vinci – Galileo laws for bodies (not their own I hope) falling freely under gravity. They stated their rules in the same basic way. I remembered these laws from watching The Mechanical Universe in high school – before taking physics from the real life version of David Goodstein three years later.

da Vinci said (or so I hear, I never met the guy) that if you fall one unit of distance in the first unit of time, you’ll fall two in the next unit, three in the one after that, then four, etc. So if you fall 5 meters in one second, in the next you’ll fall another 10 for 15 meters total.

Galileo said almost the same thing, but with odd numbers. If you fall one unit of distance in the first second, then in the second you fall three, then five more, then seven, etc. So if you again fall 5 meters in one second, in the next you’ll fall another 15, for 20 total.

Galileo was right; da Vinci wrong. But let’s not screw over our primitive-flying-device-making friend with such a cursory examination. They’re both awesome dudes, as Leonardo’s testudine counterpart would say.

Galileo was right because acceleration is constant, so the distance fallen is proportional to the square of the time. Adding Galileo’s odd numbers gives a square number. 1+3+5 = 9, for example. This is easy to see from a picture.

Each new section adds the next odd number worth of dots, and takes you to the next bigger square number when counted as a whole.

Each new section adds the next odd number worth of dots, and takes you to the next bigger square number when counted as a whole.

da Vinci, instead of the square numbers for total distance fallen, gave the triangular numbers. 1 + 2 + 3 = 6, which is triangular. This has its own picture.

According to da Vinci, each new row is how much you fall in one additional second.

According to da Vinci, each new row is how much you fall in one additional second.

da Vinci’s fub may have been in misunderstanding the relationship between speed and distance. If da Vinci’s rule had been giving the speed at the end on each second, rather than the incremental distance fallen, he’d have been right. If you’re going 10m/s after one second, you go 20m/s after two, and 30 m/s after three, etc. The problem is that you can’t find the distance traveled in a second by taking the speed at the end of that second and multiplying by time. If you do that, you get only an approximation to the correct integral, like this:

Dont worry about the numerical details.  I stole this from the internet somewhere.  da Vincis law overestimates distance fallen every second by assuming your speed at the end of the second was you speed for the entire second.

Don't worry about the numerical details. I stole this from the internet somewhere. da Vinci's law overestimates distance fallen every second by assuming your speed at the end of the second was you speed for the entire second.

It’s possible that da Vinci was actually right on about the kinematics, but that he made a mathematical error in reporting his result. I wanted to follow up on this, so I checked online to see precisely what Leonardo said. I did not succeed. Fritjof Capra’s book quotes da Vinci:

The natural motion of heavy things at each degree of its descent acquires a degree of velocity. And for this reason, such motion, as it acquires power, is represented by the figure of a pyramid.

But when I search online texts of Da Vinci’s notebooks, I can’t find this passage. I can’t find the relevant passages in my Dover copy of Richter’s translation, either. In fact, I can’t find this passage anywhere else on the entire internet, except one book that doesn’t cite the source. So I’m not sure what to make of this. da Vinci’s writings on falling bodies must be somewhere, if we know about them. But as of now I’m still uncertain. Based on the preface to my translation of the notebooks, it looks like they decided to omit some of Leonardo’s physics, since that is obviously unimportant and uninteresting to readers of his notebooks.

Let’s assume Leo had the right idea, but brain farted on the integration thing. Considering how clever Da Vinci was, his mistake is very surprising, because his law is not only empirically wrong, it is logically impossible.

To see what I mean, let’s carry out Da Vinci’s argument a little further. According to his rule, in four units of time you fall 1+2+3+4 = 10 units of distance. But the choice of how long a unit of time is was arbitrary. So let’s do it again, but consider the unit of time to be twice as long as it was previously. We’ll call these “shmunits” of time. In one shmunit of time, you have to fall three units of distance to be consistent with the first calculation. Then you fall six units of distance in the second shmunit of time, because the second has you falling twice as far as the first. After two shmunits of time, you fall a total of nine units of distance. But we already said that with the same law you fall ten units of distance! Surely if Leonardo had considered his law carefully he’d have seen this error, right?

Unless it’s not an error. What if Leonardo actually meant that you have to take the limit as your unit of time becomes infinitely short? In that case, Leonardo’s law

distance \propto t(t+1)

can simply be reduced to the correct law

distance \propto t^2.

Could this really have been what Leonardo had in mind? I think it’s possible, but not likely. The Greeks explored the basic ideas here. They approximated \pi using the method of exhaustion, and Archimedes is said to have been doing what amounted to integral calculus. If Leonardo was aware of this research, he might have stated such a law accurately. But it seems far-fetched.